Cryptology ePrint Archive: Report 2007/013

Optimised versions of the Ate and Twisted Ate Pairings

Seiichi Matsuda and Naoki Kanayama and Florian Hess and Eiji Okamoto

Abstract: The Ate pairing and the twisted Ate pairing for ordinary elliptic curves
which are generalizations of the $\eta_T$ pairing for supersingular curves have previously been proposed.
It is not necessarily the case that both pairings are faster than the Tate pairing.
In this paper we propose optimized versions of the Ate and twisted Ate pairings with the loop reduction method and show that both pairings are always at least as fast as the Tate pairing.
We also provide suitable families of elliptic curves that our optimized Ate and optimized twisted Ate pairings can be computed with half the loop length compared to the Tate pairing.